Character Complexity: A Novel Measure for Quantum Circuit Analysis

TL;DR

The paper introduces Character Complexity, a representation-theoretic measure for quantum circuits, defined as $C(U) = \frac{1}{|G|} \left(\sum_i \frac{|\chi_i(U)|^2}{d_i}\right)$, to capture circuit structure beyond gate counts. It proves fundamental properties including $0 \le C(U) \le 1$, multiplicativity on tensor products, and a general bound under composition, and shows a strong link to classical simulability: circuits with $C(U) \le c\\log(n)$ and poly$(n)$ group size $|G|$ can be efficiently simulated classically. The work also develops visualization tools based on a modified Bloch sphere to map character complexity to geometric representations, and provides empirical comparisons across circuit families (random, QFT, QAOA) to reveal scrambling dynamics. These results offer a rigorous framework for understanding the quantum-classical boundary and guide the design and optimization of quantum algorithms. Collectively, the methods connect abstract group-theoretic concepts to practical quantum computing concerns, with potential impact on algorithm development and error-correction strategies.

Abstract

In the rapidly evolving field of quantum computing, quantifying circuit complexity remains a critical challenge. This paper introduces Character Complexity, a novel measure that bridges Group-theoretic concepts with practical quantum computing concerns. By leveraging tools from representation theory, I prove several key properties of character complexity and establish a surprising connection to the classical simulability of quantum circuits. This new measure offers a fresh perspective on the complexity landscape of quantum algorithms, potentially reshaping our understanding of quantum-classical computational boundaries. I present innovative visualization methods for character complexity, providing intuitive insights into the structure of quantum circuits. The empirical results reveal intriguing scaling behaviors with respect to qubit and gate counts, opening new avenues for quantum algorithm design and optimization. This work not only contributes to the theoretical foundations of quantum complexity but also offers practical tools for the quantum computing community. As quantum hardware continues to advance, character complexity could play a crucial role in developing more efficient quantum algorithms and in exploring the fundamental limits of quantum computation.

Figures (4)

• Figure 1: Character Complexity in 10-Qubit Quantum State Space: 3D projection of the quantum state hypersphere, where each point represents a quantum state colored by its character complexity. The radius of each point is determined by $r = \sqrt{\frac{2^n - 1}{2^n}} \cdot \sqrt{1 - 2^{-C(\ket{\psi})}}$, where $n = 10$ qubits. The color gradient from blue to yellow represents increasing character complexity. The outer sphere represents the maximum possible radius for a 10-qubit state.
• Figure 2: Distribution of Character Complexity for different 6-qubit quantum circuits.
• Figure 3: Character Complexity as a function of Qubits and Gates for different 6-qubit circuit types.
• Figure 4: Character Complexity scaling with respect to gate count for different 6-qubit circuit types.

Theorems & Definitions (13)

• Definition 1
• Theorem 1: Bounds on Character Complexity
• proof
• Theorem 2: Multiplicativity under Tensor Products
• proof
• Theorem 3: General Bound on Character Complexity under Composition
• proof
• Lemma 1: Multiplicativity of Character Complexity for Abelian Groups
• proof
• Theorem 4: Strong Relation to Classical Simulation Complexity